Net changes in mass and atomic numbers from decay emissions

Steps:

• Alpha emission reduces mass number (A) by 4 and atomic number (Z) by 2 per particle.
• Beta-minus emission keeps A unchanged but increases Z by 1 per particle.
• Required net changes: ΔA = -4 (X loses 4 nucleons), ΔZ = 0 (same protons).
• Test combinations: one alpha gives ΔA = -4, ΔZ = -2; adding two betas gives ΔZ = +2, netting ΔA = -4, ΔZ = 0.

Why C is correct:

• Matches conservation laws: one alpha accounts for ΔA = -4 and ΔZ = -2; two betas compensate with ΔZ = +2, yielding exact net changes.

Why the others are wrong:

• A: Two alphas and one beta yield ΔA = -8, ΔZ = -3 (excess mass loss, wrong Z).
• B: One alpha and one beta yield ΔA = -4, ΔZ = -1 (correct A but insufficient Z increase).
• D: Identical to A, yields ΔA = -8, ΔZ = -3 (excess mass loss, wrong Z).

Final answer: C